Error in rational.toDouble

[harshad-deo]

The toDouble method on Rational instances returns incorrectly for certain inputs. To reproduce,

import spire.math.Rational

def bld(numStr: String, denomStr: String): (Double, Double) = {
  val num = BigInt(numStr)
  val denom = BigInt(denomStr)

  val expected = (BigDecimal(num) / BigDecimal(denom)).toDouble
  val actual = Rational(num, denom).toDouble

  (expected, actual)
}

println(
  bld(
    "179709322496314735297382168733437804616092100361307360024524978382010425322923559",
    "179709322496314735297382168733483632988275016201074155389238170380281177201180672"
  ))
println(
  bld(
    "-1257965257474203147081675181125850088132776039278857905825424935347911159685302509",
    "1257965257474203147081675181134385430917925113407519087724667192661968240408264704"
  ))
println(bld(
  "3618502788666131106986593281458997943466459808578940660043329292438660832363",
  "3618502788666131106986593281521497120414687020801267626233049500247285301248"
))

spire version: 0.14.1 scala version: 2.12.4 javac: 1.8.0_152 jvm: Java HotSpot(TM) 64-Bit Server VM (build 25.152-b16, mixed mode) OS: Ubuntu 16.04

[denisrosset]

That's probably due to the algorithm used in Rational.toDouble.

1. We should have principled approximations, i.e. laws that such conversions such obey. For example, decide to map to the nearest float.

2. Help is needed to turn this code in a test case. We should avoid using BigDecimal division as a comparison (so much can go wrong with the approximation in use). However, it is clear that the ratio of very close (big) integers cannot be approximated by 0.5.

[kschwarz1116]

This and #870 look like the same issue (or at least the same class of issue). Arguably they should be tackled together.